#493203

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#20476
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,12,7,14,28,41,8,13,29,40,30,21,35,51)(3,15,19,44,20,45,49,63,18,9,31,33,48,34,58,59)(4,10,22,5,11,25,38,27,23,36,52,26,39,50,60,56)(16,42,46,54,37,61,62,17,43,32,57,24,53,55,64,47);(1,3)(2,9,13,15)(4,16)(5,24,26,54)(6,19)(7,20)(8,18)(10,32,36,42)(11,37)(12,33,40,44)(14,34,21,45)(17,27,47,56)(22,46)(23,43)(25,55,50,61)(28,49)(29,31)(30,48)(35,58)(38,62)(39,53)(41,59,51,63)(52,57)(60,64);(1,4,6,22,7,11,28,38,8,23,29,52,30,39,35,60)(2,10,12,5,14,25,41,27,13,36,40,26,21,50,51,56)(3,16,19,46,20,37,49,62,18,43,31,57,48,53,58,64)(9,32,33,24,34,55,59,47,15,42,44,54,45,61,63,17);(1,5)(2,11)(3,17,18,47)(4,21)(6,25)(7,27)(8,26)(9,16,15,43)(10,35)(12,38)(13,39)(14,23)(19,32,31,42)(20,24,48,54)(22,51)(28,36)(29,50)(30,56)(33,46,44,57)(34,37,45,53)(40,60)(41,52)(49,55,58,61)(59,62,63,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,h^2=i,b^a=be,g^a=gi,d^b=dh,e^b=ei,d^c=di,g^d=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224955, #72224961, #72224962, …
maximal normal subgroup of	#72224955, #72224961, #72224962, …
maximal quotient of	C₂×#493203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400399);